2.1 Financial network and stock market synchronization

The methodologies to study the inter-market and intra-market interconnectedness study the co-movement between the returns of the assets, caused by the behavior of the investor (^{Barberis & Thaler, 2005}) (^{Green & Hwang, 2009}) and macroeconomic factors (^{Cohen & Frazzini, 2008}). Among the most prominent methodologies are the unit roots and cointegration test, autoregression vector models, correlation-based test (^{Forbes, & Rigobon, 2002}), causality test (^{Billio, Caporin, Frattarolo & Pelizzon, 2021}), GARCH models multivariate (^{Engle, 2002}), and variance decomposition models (^{Diebold & Yilmaz, 2009}).

Due to their internationalization and increase in investment alternatives, the increasing complexity of financial markets has motivated the development of methodologies that study correlations and volatilities and incorporate complexity while presenting the phenomenon in a parsimonious way. ^{Onnela, Chakraborti, Kaski, & Kertesz (2003)} argue that the minimal spanning tree length (MSTL) simply the market's complexity setting clusters between world stock markets (^{Eryiği & Eryiğit, 2009}), correlations between markets and clusters (^{Gao & Mei, 2019}), and identifying between centrality and peripheral markets (^{Zhao, Li & Cai, 2016}).

Additionally, MSTL increases during the financial crisis, turning it into an efficient crisis indicator. In other words, in crises, MSTL contracts due to high asset correlation (^{Wang Xie & Stanley, 2018}) which leads to the costs per position in an investment increasing, making diversification difficult (^{Coelho, Gilmore, Lucey Richmond & Hutzler, 2007}).

Finally, MSTL is very efficient in demonstrating the phenomenon of market synchronization, significant for financial stability and investors' diversification strategies. New literature has shown that this correlation is not constant over time and varies significantly from one period to another (^{Magner, Lavin, Valle & Hardy, 2020}). This phenomenon is known as stock market synchronization, and it represents the sum of the correlations between the assets that make up a market.

2.2. Volatility forecasting

^{McAleer, & Medeiros (2008)} study models that forecast the assets’ volatility to find optimal investment strategies and anticipate market movements, reducing economic losses from crisis. However, volatility is not directly observable despite being in a latent state (^{Andersen, Bollerslev & Meddahi, 2005}) (^{Koopman, Jungbacker & Hol, 2005}), using the realized volatility as a real volatility proxy that is consistent estimates of the true (latent) integrated volatility, becoming a benchmark practical.

The volatility is essential to asset pricing, derivatives, portfolio selection, risk management, and hedge (^{Wang, 2019}). Therefore, its forecast becomes essential for decision-making in destabilizing scenarios for the system, increasing risk (^{Magner, Lavin, Valle & Hardy, 2020}). In a standard scenario, the lower price fluctuation causes greater control in risk management; however, peak moments generate great uncertainty that is quickly transferred to prices, being able to obtain a better forecast probability for periods after these. "Jumps" (stochastic processes that have discrete movements, with random arrivals and many continuous movements) (^{Clement & Liao, 2017}).

Other methodologies include the VIX as an essential risk indicator; a large VIX affects the international market more significantly instead of the smaller original VIX, finding an improvement in the forecast for the first case and demonstrating that it impacts significantly in the volatility of a step forward (^{Wang, 2019}). Additionally, under structural ruptures, linear combination methods outperform non-linear methods when improving risk prediction using different window sizes. Even with all the discoveries and models used, volatility is still a complicated indicator to forecast.

3.1 Returns and realized variance

We use daily closing price p to estimate the daily logarithmic return R_i,t of day t for asset i where, Ri,t=ln (Pi,t) -ln(Pi,t-1). As ^{Andersen, Bollerslev, Diebold, & Ebens (2001)} we obtain the realized volatility for each stock through the daily return of the asset:

RV is the realized volatility, and R represents the logarithmic return of asset i on day t of month t for days C_T of the respective month T.

3.2 Minimum spanning tree length (MSTL)

The Minimum spanning tree is based on the correlation of the assets' daily returns under study (^{Mantegna, 1999}). In this research, we use the Pearson correlation in time-variable series (Benesty et al., 2009), where i,j quantifies the correlation relationship between assets' return on edge i and j, , when i and j are stocks, representing a Pearson correlation matrix NxN (where N represents the number of stocks). Finally, we convert the correlation in distances (d) (equation 2) for each edge of assets i,j evaluated in the month T.

The distance (d), in this way, means the continuous quantitative relationship between the edge i,j, , when i and j are stocks, with limit values between 0 ≤ d_i,j ≤ 2. With these distances, we calculate an adjacency matrix weighted network of N(N-1)/2 elements used quantitatively as distance, where, in case of representing a higher correlation, the distance tends to 0. In contrast, when representing a lower correlation, the distance tends to 2.

From the adjacency matrix, we extract the smallest distance of each edge i,j (^{Prim, 1957}), when i and j are stocks, and estimate the MST representing the minimum distance for the connection between all network assets (^{Mantegna, 1999}). Finally, we quantify the euclidean distance between each edge i,j and obtain the MSTL (equation 3) for 162 monthly (T).

The minimum distance's length between all the assets studied is represented as a weighted euclidean variable of the network for each month, where the length is reflected numerically between 0 ≤ MSTL(T) ≤ 2*(N-1). The distance length MSTL tends to its Minimum when the individual distance tends to zero, while when the individual distance tends to 2, the MSTL tends to its maximum so that both demonstrative variables present a negative correlation. ^{Onnela, Chakraborti, Kaski, & Kertesz (2003)} argue that the minimum spanning tree length (MSTL) is simply the complexity of the market that establishes groupings between world stock markets (^{Eryiğit, & Eryiğit, 2009}), correlations between markets and groupings (^{Gao & Mei, 2019}). What makes it a complete indicator compared to others that do not measure the relationship between assets.

The MSTL and realized volatility series of assets used in the estimates follow the stationarity expressed by ^{Banerjee, Doran & Peterson (2007)} and ^{Perron (1988)} without persistence, varying periodically in the function of economic cycles.

3.3 Forecast test

Autoregressive models have commonly explained asset volatility (^{Corsi, 2009}). For this research, we add information from the systemic network as an explanatory variable of each shares' volatility by representing the monthly MSTL, each residual of the models complies with assumptions of normality, homoscedasticity and no auto-correlation.

To explain the monthly realized volatility of the shares and subsequently evaluate the influence of the explanatory variable under study (MSTL), we use an AR (p) model within the sample that includes 6 temporary lags of the asset volatility plus a lag temporary of MSTL (Table 1, panel a). In the sample, we estimate with all the observations, having as main objective to validate our alternative hypothesis H1: βi≠0 the existence of Granger causality (^{Granger, 1969}) in the explanatory variable (MSTL) included in our models

Out of sample, we estimate through recursive windows considering a p / r = 0.4, p / r = 1, p / r = 2, obtaining results through the ENCNEW test (^{Clark & McCracken, 2001}), for this, we generate a Benchmark model (Table 1, panel c) and compare the core model for each stock (table 1, panel b). For each of the 26 companies, the null hypothesis H1: βi≠0, will be validated if the MSTL generates a decrease in the forecast error when incorporated into the benchmark model, validating the effectiveness of the inclusion of our variable. For the estimation, we used a HAC standard error methodology for stationary covariance processes in time series, correcting the deviations of homoscedasticity and autocorrelation of errors produced by the long-term variance (^{Newey, & West, 1987}) (^{Newey, & West, 1994}).

3.4 VAR and Forecasting Error Variance Decomposition

We performed a VAR to test the causal relationship between two stationary variables. This method has proven to be especially useful to describe the dynamic behavior of economic, financial time series, and forecast evaluations.

We estimate a VAR model of simultaneous equations using two lags of the MSTL and the volatility realized for each share (See Table 1, Panel D), complying with the assumptions of normality, homoscedasticity and no auto-correlation for the residuals. The main purpose of this estimation is to validate our alternative and independent hypotheses of the existence of Granger causality in the explanatory variable (MSTL) with both lags included in our models with estimation by ordinary least squares.

Additionally, we generate an impulse in the MSTL to measure this shock's intensity in the realized volatility of assets and in the network itself. In case these impulses can explain future volatility spills, we interpret them as significant drivers. In this way, decomposing the variance of the forecast error (^{Diebold, & Yılmaz, 2014}), we measure how much the innovation contributes to the variance of the total forecast error from H months ahead for each realized volatility for the quarters of the coming year (H = 1, H = 3, H = 6, H = 9, H = 12) since the shock occurred.

Where ASt,H represents the sum of variance percentages of the forecast error from H months ahead, using data from month t. AS shows the impulse response and ah,(i,MSTL)2 represents the contribution to the variance of the error to forecast assets i due to changes in MSTL.

4.1 Synchronization stock market in Chile

Table 2 reports the results of descriptive statistics and unit root test calculated for each of the equity assets, including information for the total study period and the MSTL synchronization index during different sub-periods of the sample.

The results show that energy companies (Copec, Enelam, Colbún, and ECL), banking (Bsantander, Chile, BCI), and retail (Cencosud, Falabella, and Ripley) present less centrality, reducing their probability of failure in high times and its similarity in behavior with other assets. This phenomenon translates into an opportunity for diversification optimization (^{Peralta & Zareei, 2016}). On the contrary, assets in the beverage sector (Conchatoro and Andinab), and other companies as Security, Aguasa, IAM, and Vapores, present a greater centrality, being the primary source and receiver of information transfer in the market (^{Sensoy, Nguyen, Rostom & Hacihasanoglu, 2019}).

When observing the MSTL synchronization, skewness tends to approach zero in the pre-crisis and crisis periods, implying the network's Gaussian distribution (^{Coelho, Gilmore, Lucey, Richmond & Hutzler, 2007}). Also, the average value of the network during these periods tends to decrease concerning the post-crisis period and the total period, which indicates that the correlation between assets increases, implying that the market synchronization also does the same. On the other hand, there is an increase in the data's standard deviation since the uncertainty and economic events during these periods cause unexpected returns on assets.

In the post-crisis period, we observe that assets tend to increase their distance compared to the pre-crisis and crisis periods, showing less market synchronization and decreased data deviation with values closer to those reflected in the entire network.

4.2 In-Sample Analysis

Table 3 reports the models reported in Table 1, panel A. We consider monthly frequencies and use HAC standard errors according to ^{Newey & West (1987)} and ^{Newey, & West (1994)}. The results show that the MSTL coefficients are negative and statistically significant for 15 of the 26 actions studied, demonstrating a greater than 50% predictive power. This result means that an expansion (contraction) in the asset correlation network predicts a decrease (increase) in volatility realized in the following month by more than 50% of the market. This negative lagged coefficient is consistent with the research of ^{Magner, Lavin Valle & Hardy (2020)}, which found a negative and lagged relationship between the variation of the MSTL and PMFGL and the realized volatility of 11 stock market indices worldwide.

The assets whose realized volatility has a statistical significance with the delayed synchronization of the market belong to the retail industries (Cencosud, Falabella, Ripley), energy (Copec, Enelam, Colbun, ECL), banking sector (BSantander, Chile, BCI), beverage (Concha y Toro, AndinaB), utilities (IAM), tech-companies (Sonda), and real estate (Salfacorp). Industrial sectors, telecommunications, other financial services, and raw materials extraction did not present a statistical relationship with the lagged synchronization index.

Regarding the retail industry, Table 3 shows that the realized volatility of the Cencosud stock (beta = -1.30.E-03; p = 1.09.E-02), Falabella (beta = -1.19.E-03; p = 3.11.E-02), and Ripley (beta = -1.44.E-03; p = 2.00.E-04) turned out to be statistically significant, increasing when the MSTL of the network decreased in the previous month. In other words, an increase in synchronization is a preview of an increase in realized volatility of companies in the Retail sector.

Respect to the energy industry, Table 3 shows that the realized volatility of the stock is inversely related to the synchronization of the stock market lagged in one month. Thus, Copec (beta = -6.26.E-04; p = 5.94.E-02), Enelam (beta = -6.37.E-04; p = 5.85.E-02), Colbun (beta = -7.53.E -04; p = 1.37.E-02), and ECL (beta = -1.06.E-03; p = 6.62.E-02) show an increase in their volatility realized after a decrease in the MSTL of the network of assets.

Stocks' realized volatility in the banking sector also showed a negative and statistically significant relationship with the lagging MSTL of the network. Within this group, in Table 3 we see that Banco Santander (beta = -9.65.E-04; p = 1.31.E-02), Banco Chile (beta = -1.04.E-03; p = 4.37.E- 02), and BCI (beta = -7.67.E-04; p = 7.90.E-03) turned out to be statistically significant, increasing when the MSTL of the network decreased in the previous month. The only exception was the realized volatility of ItauCorp (beta = -5.62.E-04; p = 1.82.E-01) and Security (beta = -6.27.E-04; p = 1.14.E-01) which did not showed a statistically significant relationship with market timing.

Other companies show a negative and statistically significant relationship between their realized volatility and the lagged MSTL of the market. Table 3 shows that such companies are from the drinking sector: Concha y Toro (beta = -8.36.E-04; p = 3.52.E-02) and AndinaB (beta = -4.89.E-04; p = 9.28.E- 02); utilities sector: IAM (beta = -5.89.E-04; p = 8.30.E-02); Sonda technology sector (beta = -1.33.E-03; p = 5.20.E-03), and Salfacorp construction sector (beta = -2.40.E-03; p = 1.05.E-02).

Additionally, we note features derived from the in-the-sample model. First, we note a strong autocorrelation in the realized volatility of the stocks studied. Table 3 shows that we noticed 25 out of a total of 26 companies present a positive and statistically significant constant and positive statistical significance in any of the 6 lags of realized volatility.

4.3 Out-of-Sample

Tables 4 show the ENCNEW test results (^{Clark & McCracken, 2001}) in out-of-sample exercise for all stocks. The results of table 4 correspond to the statistical difference between the core model reported in table 1 panel B versus the benchmark models presented in Table 1 panel C when the proportion of the sample to make the forecast corresponds to a 40%, 50%, and 67% (P / R = 0.4, P / R = 1, and P / R = 2, respectively)

In Table 4, we use an AR (6) model described in Table 1, denoting an autoregressive process of order 6 for the realized volatility of each stock included in the study. For the case, P / R: 0.4, 17 of the 26 stocks studied presented a statistically significant difference. We interpret these results as the core model (See Table 1, panel B) has better predictive capacity than the benchmark model (See Table 1, panel C). Of the 17 actions that present statistical significance, 14 reject the null hypothesis with a probability of less than 1%. In contrast, the remaining 3 companies reject the hypothesis with a probability of less than 5%.

When performing the out-of-sample tests, increasing the estimation window to 50% (P / R: 1), 13 companies reject the null hypothesis with a probability less than 10% (See Table 4 column 3); while the same test with an estimation window of 67% (P / R: 2) indicates that 5 companies reject the null hypothesis with a probability less than 10%.

The industries that showed the best forecasting performance between the lagged MSTL and the stock's realized volatility are retail, banking, energy, beverage, technology, and utilities, consistent with the results obtained in the out-of-sample tests.

4.4 VAR and Forecast Error Variance Decomposition Analysis

Tables 5 and 6 show the VAR (2) results using the MSTL and the realized volatility of each asset. We selected the AR (2) using the methodology for selecting the appropriate delay (^{Pfaff, 2008}). This analysis aims to study Granger Causality; the results show partial causality in one direction for 10 of the 26 companies studied. This result means that the lagged MSTL presents a statistical significance on the stock's realized volatility for the following period (See Table 5). Additionally, we do not observe causality in the opposite direction. In other words, the lagged realized volatility of financial assets is not related to the timing of the market in the next period.

Another interesting result is the strong causality verified in the stock market synchronization phenomenon. Table 6 shows that the first lag of the MSTL is statistically significantly related to the MSTL of the following period. The persistence in the MSTL time series, which we interpret as the synchronization of the stock market causing a significant part of the synchronization of the same market in the following period.

Finally, we organize an orthogonalized disturbance that contributes to the mean squared error (MSE) in the H-periods-ahead forecasts (See Table 7). Our objective is to study the percentage contribution to each asset's realized volatility (Table 7) and the MSTL market synchronization (Table 8). The results show that the most significant proportion of each asset's variance is caused by the lagged realized volatility of the asset, and to a lesser extent, by the market synchronization (Table 7).

Regarding market timing, Table 8 shows that the lag explains the vast majority of the variance in market timing and that the realized volatility of each asset explains a minimal (almost negligible) portion.